Pythagoras' theorem calculator (find any missing side)
Enter any two sides of a right-angled triangle and this calculator finds the third. It shows whether you are adding or subtracting the squares, gives the exact surd answer for non-calculator papers, and rounds sensibly for calculator papers. It also returns the area, perimeter and the two acute angles.
Enter two sides — leave the missing one blank
a and b are the two shorter sides; c is the hypotenuse, opposite the right angle.
c = 5
Sides: a = 3, b = 4, c = 5
Area: 6 · Perimeter: 12
Angles: 36.9° and 53.1° (plus the 90° right angle)
Working
- c² = a² + b² = 9 + 16 = 25
- c = √25 = 5
The method behind it
- Hypotenuse: c = √(a² + b²) — add the squares.
- Shorter side: a = √(c² − b²) — subtract the squares.
- Simplify the root for an exact surd answer, or round to 3 s.f. on calculator papers.
How to use this tool
1. Identify the hypotenuse
The hypotenuse is the longest side, always opposite the right angle. Everything in Pythagoras depends on labelling this correctly.
2. Enter the two sides you know
Fill in the two boxes you have and leave the missing side blank. Units do not matter as long as they match.
3. Read the working
If you are finding the hypotenuse you add the squares; if you are finding a shorter side you subtract. The steps show which case applies to your triangle.
Frequently asked questions
What is Pythagoras' theorem?
In a right-angled triangle, a² + b² = c², where c is the hypotenuse and a and b are the two shorter sides. It only works for right-angled triangles — for other triangles use the cosine rule.
How do you find a shorter side with Pythagoras?
Rearrange to a² = c² − b², so the shorter side is √(c² − b²). Subtract rather than add whenever the hypotenuse is one of the sides you already know.
Should I leave the answer as a surd?
On non-calculator papers, yes — √52 simplified to 2√13 is the exact answer and earns the accuracy mark. On calculator papers, round to 3 significant figures or to whatever accuracy the question states.
How do I check my triangle is right-angled?
Square all three sides. If the two smaller squares add up to the largest square, the triangle is right-angled (the converse of Pythagoras). If they do not, it is not right-angled.
Practise this properly, not just once
A solver gives you the answer; exam marks come from the working. MathsGradeUp pairs exam-board questions with step-by-step feedback and revision notes on the exact topics you keep dropping marks on.